Question: Simplify; express your answer in exponential form. Assume $q\neq 0, a\neq 0$. $\dfrac{{(qa^{3})^{-2}}}{{q^{-5}a^{2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(qa^{3})^{-2} = (q)^{-2}(a^{3})^{-2}}$ On the left, we have ${q}$ to the exponent ${-2}$ . Now ${1 \times -2 = -2}$ , so ${(q)^{-2} = q^{-2}}$ Apply the ideas above to simplify the equation. $\dfrac{{(qa^{3})^{-2}}}{{q^{-5}a^{2}}} = \dfrac{{q^{-2}a^{-6}}}{{q^{-5}a^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-2}a^{-6}}}{{q^{-5}a^{2}}} = \dfrac{{q^{-2}}}{{q^{-5}}} \cdot \dfrac{{a^{-6}}}{{a^{2}}} = q^{{-2} - {(-5)}} \cdot a^{{-6} - {2}} = q^{3}a^{-8}$